Conservation laws (This is episode 32)
The assumption of a fluid continuum allows problems in aerodynamics to be solved using fluid dynamics conservation laws. Three conservation principles are used:
Conservation of mass
Conservation of mass requires that mass is neither created nor destroyed within a flow; the mathematical formulation of this principle is known as the mass continuity equation.
Conservation of momentum
The mathematical formulation of this principle can be considered an application of Newton's Second Law. Momentum within a flow is only changed by external forces, which may include both surface forces, such as viscous (frictional) forces, and body forces, such as weight. The momentum conservation principle may be expressed as either a vector equation or separated into a set of three scalar equations (x,y,z components).
Conservation of energy
The energy conservation equation states that energy is neither created nor destroyed within a flow, and that any addition or subtraction of energy to a volume in the flow is caused by heat transfer, or by work into and out of the region of interest.
Together, these equations are known as the Navier-Stokes equations, although some authors define the term to only include the momentum equation(s). The Navier-Stokes equations have no known analytical solution and are solved in modern aerodynamics using computational techniques. Because computational methods using high speed computers were not historically available and the high computational cost of solving these complex equations now that they are available, simplifications of the Navier-Stokes equations have been and continue to be employed. The Euler equations are a set of similar conservation equations which neglect viscosity and may be used in cases where the effect of viscosity is expected to be small. Further simplifications lead to Laplace's equation and potential flow theory. Additionally, Bernoulli's equation is a solution in one dimension to both the momentum and energy conservation equations.
The ideal gas law or another such equation of state is often used in conjunction with these equations to form a determined system that allows the solution for the unknown variables.
